MinT - Best Known (245−85, 245, s)-Nets in Base 3

Best Known (245−85, 245, s)-Nets in Base 3

(245−85, 245, 162)-Net over F₃ — Constructive and digital

Digital (160, 245, 162)-net over F₃, using

t-expansion [i] based on digital (157, 245, 162)-net over F₃, using
- 5 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
    - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
        F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(245−85, 245, 336)-Net over F₃ — Digital

Digital (160, 245, 336)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−85, 245, 4840)-Net in Base 3 — Upper bound on s

There is no (160, 245, 4841)-net in base 3, because

1 times m-reduction [i] would yield (160, 244, 4841)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 261 823443 896663 142909 237857 713615 885631 435422 328194 756043 562735 955943 328329 005553 086262 208436 889170 723989 560379 909945 > 3²⁴⁴ [i]